Continuous Bernoulli priors and certainty ($P(X_{n+1}=1\mid X_1=\cdots =X_n=1)=\frac{n+1}{n+2}$)

With a uniform prior distribution on a Bernoulli chance $\theta$, $n$ successes yield posterior density $(n+1){\theta }^n$. Integration gives next-trial success probability $(n+1)/(n+2)$, but the singleton $\theta =1$ retains probability zero. More generally, ordinary conditioning with a finite integrable likelihood and positive finite normalization constant preserves absolute continuity, so a continuous prior does not acquire a new atom at certainty from a finite sample.